Markov blanket

In statistics and machine learning, when one wants to infer a random variable with a set of variables, usually a subset is enough, and other variables are useless. Such a subset that contains all the useful information is called a Markov blanket. If a Markov blanket is minimal, meaning that it cannot drop any variable without losing information, it is called a Markov boundary. Identifying a Markov blanket or a Markov boundary helps to extract useful features. The terms of Markov blanket and Markov boundary were coined by Judea Pearl in 1988.^[1] A Markov blanket can be constituted by a set of Markov chains.

Markov blanket[edit]

A Markov blanket of a random variable ${\displaystyle Y}$ in a random variable set ${\displaystyle {\mathcal {S}}=\{X_{1},\ldots ,X_{n}\}}$ is any subset ${\displaystyle {\mathcal {S}}_{1}}$ of ${\displaystyle {\mathcal {S}}}$, conditioned on which other variables are independent with ${\displaystyle Y}$:

$${\displaystyle Y\perp \!\!\!\perp {\mathcal {S}}\backslash {\mathcal {S}}_{1}\mid {\mathcal {S}}_{1}.}$$

It means that ${\displaystyle {\mathcal {S}}_{1}}$ contains at least all the information one needs to infer ${\displaystyle Y}$, where the variables in ${\displaystyle {\mathcal {S}}\backslash {\mathcal {S}}_{1}}$ are redundant.

In general, a given Markov blanket is not unique. Any set in ${\displaystyle {\mathcal {S}}}$ that contains a Markov blanket is also a Markov blanket itself. Specifically, ${\displaystyle {\mathcal {S}}}$ is a Markov blanket of ${\displaystyle Y}$ in ${\displaystyle {\mathcal {S}}}$.

Markov boundary[edit]

A Markov boundary of ${\displaystyle Y}$ in ${\displaystyle {\mathcal {S}}}$ is a subset ${\displaystyle {\mathcal {S}}_{2}}$ of ${\displaystyle {\mathcal {S}}}$, such that ${\displaystyle {\mathcal {S}}_{2}}$ itself is a Markov blanket of ${\displaystyle Y}$, but any proper subset of ${\displaystyle {\mathcal {S}}_{2}}$ is not a Markov blanket of ${\displaystyle Y}$. In other words, a Markov boundary is a minimal Markov blanket.

The Markov boundary of a node ${\displaystyle A}$ in a Bayesian network is the set of nodes composed of ${\displaystyle A}$'s parents, ${\displaystyle A}$'s children, and ${\displaystyle A}$'s children's other parents. In a Markov random field, the Markov boundary for a node is the set of its neighboring nodes. In a dependency network, the Markov boundary for a node is the set of its parents.

Uniqueness of Markov boundary[edit]

The Markov boundary always exists. Under some mild conditions, the Markov boundary is unique. However, for most practical and theoretical scenarios multiple Markov boundaries may provide alternative solutions.^[2] When there are multiple Markov boundaries, quantities measuring causal effect could fail.^[3]

See also[edit]

• Andrey Markov
• Free energy minimisation
• Moral graph
• Separation of concerns
• Causality
• Causal inference

Notes[edit]